How to identify family of function from given property

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A={$f:R\to R$}

with following properties

f has derivative of every order

$\forall x,y\in R$ $f(x+y)-f(y-x)=2xf'(y)$

then I have to find whole family

Till now I just able to show $f(x)-f(-x)/2x=2xf'(0)$

I wanted to solve this problem.

Please provide just hint..

Thanks in Advanced